Score:1

Showing that $F'(k, x) := F(F(k, 0^{n}), x)$ is a PRF

jp flag

I wanted to do some practice on security reduction proofs, and I am stumped on this one from the Boneh-Shoup book.

If $F(k, x)$ is a secure PRF, then show that $F'(k, x) := F(F(k, 0^{n}), x)$ is a secure PRF.

What I have so far is:

Suppose $F'$ is insecure, with a distingisher $D'$. This means that $F$ is also insecure, with a distinguisher $D$. I will now show construct $D$ using $D'$.

  1. $D$ receives a key, $k$.
  2. $D$ starts running $D'$.
  3. Whenever $D'$ queries its oracle on a message $x \leftarrow \lbrace0,1\rbrace^{n}$, give $x$ to $D$, compute $y:= O(x)$, where $O$ is $D$'s oracle. Then send $F(F(k, y), x)$ to $D'$.
  4. Output whatever $D'$ outputs.

This means that:

Pr[$D'^{F'}(1^{n}) = 1] =$ Pr[$D^{F}(1^{n}) = 1]$ and Pr[$D'^{r}(1^{n}) = 1] =$ Pr[$D^{r}(1^{n}) = 1]$, where $r$ is a random function. As well,

$|$Pr$[D^{F}(1^{n}) = 1] - Pr[D^{r}(1^{n}) = 1]| > $ negl($n$)

by assumption. However, since $F$ is a PRF, this is a contradiction, so $F'$ is a PRF. $\square$

Does this proof make sense? I have a feeling I messed up defining $D$, but I'm not sure. Thanks for any help!

Fractalice avatar
in flag
Why does $D$ receive a key? You have $F(F(k,y),x)$ sent to $D'$ (with $y=F(k, x)$), while it distinguishes the shape $F(F(k,0),x)$. Can you fix that? You also need to argue why what you send to $D'$ is "like random" when $F(k, .)$ is random.
ness64 avatar
jp flag
@Fractalice Hmm, so instead of $y=O(x)$, would it just be $y=O(0^{n})$, and then $D$ sends $F(y,x)$ to $D'$? I guess instead of getting a tag $D$ can compute $y = O(0^{n})$ in step 1.
mangohost

Post an answer

Most people don’t grasp that asking a lot of questions unlocks learning and improves interpersonal bonding. In Alison’s studies, for example, though people could accurately recall how many questions had been asked in their conversations, they didn’t intuit the link between questions and liking. Across four studies, in which participants were engaged in conversations themselves or read transcripts of others’ conversations, people tended not to realize that question asking would influence—or had influenced—the level of amity between the conversationalists.